Before deriving the key estimates of \(\Vert \nabla \mathbf {u}\Vert _{L^{\infty }(0, T; L^2)}\)  and \(\Vert \nabla \mathbf {H}\Vert _{L^{\infty }(0, T; L^2)}\) , we insert an important estimate on magnetic field \(\mathbf {H}\)  initiated by He and Xin [1]}, which will be stated in the following lemma.
